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>> In this module we're going to briefly discuss the Black-Scholes formula.

Â The Black-Scholes formula is of great significance.

Â Its used awful lot in industry, and indeed we can view the binomial model as an

Â approximation to the Black-Scholes formula.

Â Black and Scholes assumed a continuously compounded interest rate of R, so that $1

Â invested in the cash account at time 0 would be worth E to the R-T dollars at

Â time T. They also assumed Geometric Brownian

Â motion dynamics for the stock price, so that the stock price at time little t, is

Â equal to the initial stock price times the exponential at this quantity here where wt

Â is a standard Brownian motion. And for those of you who are interested we

Â have recorded separate modules on both Brownian motion and Geometric Brownian

Â motion; These modules are available on the course website.

Â They also assumed that the stock pays a dividend yield of c.

Â They assumed continuous trading with no transactions cost.

Â And they also assumed that short selling was allowed.

Â So here are some sample paths of Geometric Brownian motion.

Â They look quite similar to sample paths of Brownian Motion actually.

Â You can see that the stock price never jumps here.

Â So at no point you see the stock price, say jumping from here down to here.

Â So that's a property of Brownian Motion and Geometric Brownian Motion.

Â The sample paths are actually continues. Given the assumptions that Black and

Â Scholes made and that we listed two slides ago, they succeeded in deriving the price

Â for European coal option, with strike k and maturity t.

Â It is given to us by this quantity here. It's a somewhat complicated looking

Â formula, but it's easy to code up, and it is used everywhere in industry today.

Â The interesting thing to note about this formula is that, mu, which was the drift

Â of the Geometric Brownian motion two slides ago.

Â So recall, we assumed a drift of mu here. Well, if you notice, over in the

Â Black-Scholes formula, mu does not appear anywhere.

Â And this is similar to the fact that p, did not appear in the option pricing

Â formulas we derived in the context of the binomial model.

Â We'll return to that in a moment. So European put option prices, p zero, can

Â then be calculated from put call parity. So once we have the call option price up

Â here, we can price put options. In other words, we can derive P0 using put

Â call parity here. And this is the version of put call parity

Â that applies when we have a dividend yield.

Â Black and Scholes obtained their formula using a similar replicating strategy

Â argument to the one that we used for the binomial model.

Â In fact you can show that under the Black-Scholes Geometric Brownian motion

Â model, that we can write the price of the option C0, as being the expected

Â discounted pay off of the option using risk-neutral probabilities Q.

Â This is exactly the same formula we have for the binomial model.

Â In this case however, in the continuous time model of Geometric Brownian motion,

Â under Q we assume that the stock price is given to us by this here and the only

Â difference between this expression for St and the expression we have for St at

Â couple of slides ago. Which was mu minus sigma squared over 2

Â times T, plus sigma Wt is that we now have a factor r minus c appearing here, which

Â we don't have down here. And the true drift of the Geometric

Â Brownian motion mu, no longer appears in the option pricing formula.

Â This is exactly analogous to the fact that we use the risk mutual probabilities Q and

Â not the true probabilities P when we are pricing options in the binomial model.

Â So in fact, if you evaluate this rate this expectation, assuming St is equal to this,

Â you'll get the Black-Scholes formula. And for those of you who are interested,

Â it's actually not very difficult to do this, it involves an integral of a log

Â normal distribution. St here will have a log normal

Â distribution, so one can actually evaluate this, do some integration and actually get

Â the Black-Scholes formula that we showed on the previous slide.

Â The Black-Scholes formula is used a great deal in industry, in fact it is the way in

Â which option prices are actually quoted by industry practitioners.

Â The binomial model is often used as an approximation to the Black-Scholes model,

Â in which case one needs to translate the Black-Scholes parameters R sigma and so

Â on, into R familiar binomial model parameters.

Â This is often referred to as the calibration of a Binomial Model, so

Â suppose we are given some Black Scholes parameters we have R and we have Sigma,

Â and if you notice over here that's all we need, we have R we have Sigma, we also

Â have c of course. I'm going to see how to calibrate these

Â and rewrite these parameters in our binomial model.

Â So what we will do is, we will write rn and now we're going to have subscript n

Â here to denote the fact that these parameters in the binomial model, will

Â depend on the number of periods that we're using in the binomial model.

Â So recall, if t is the maturity, of the option, then t is equal to n times delta-t

Â where delta-t is the length of a period, In the binomial model.

Â Okay, so we're going to assume that rn is equal to e to the rt over n, Where n is

Â the number of periods, as we said. We'll assume that rn minus cn.

Â So this is rn minus c in the binomial model will be equal to e to the r minus c

Â times t over n. And a simple first order taylor expansion

Â will tell you that this is approximately equal to 1 plus r t over n minus c t over

Â n. So this of course, Is like our R factor,

Â and this is our C factor, in the binomial model.

Â We'll set U-n equal to this, and D n equal to 1 over u-n.

Â And now we can price European and American options, and futures, and so on, as

Â before, in the binomial model using these parameters.

Â The risk mutual para-, probabilities, will be calculated as, q subscript n.

Â Again, recognize the dependence of our paremeters on n, the number of periods.

Â So q subscript n will be equal to this here.

Â And using this approximation, we can see that this is approximately equal to our rn

Â minus cn minus dn, divided by un minus dn. So this the representation of the risk

Â mutual probabilities that we saw before in the binomial model.

Â We're actually going to use this in our binomial model now, when we're deriving

Â our parameters from a Geometric Brownian motion and the Black-Scholes Formula.

Â So our spreadsheet, will actually calculate binomial parameters, in this

Â way. I mentioned at the beginning of the

Â module, that the binomial model can be viewed as an approximation to geometric

Â Brownian motion, This is true, as delta t, the length of a period Goes to zero.

Â I'm just going to spend a couple of moments describing how you might go about

Â showing this. We certainly won't go through all the

Â calculations, but I'll do the first couple of steps of these calculations.

Â Recall that we can calculate European option prices with strike k, according to

Â this expression here. So this is the expression we had in our

Â binomial model. C0 equals the discount factor times the

Â expected pay-off of the option, using the risk neutral probabilities Q.

Â Well, if you recall, we also saw that we can write this expression, as we have

Â written here, so these are the binomial probabilities.

Â Okay, so this here, is equal to the probability inner binomial model of j

Â upmoves, and n minus j down-moves. And this then is St, the terminal stock

Â price, after the j up-moves and n minus j down-moves.

Â So therefore this here, is the expected value of the pay-off of the option.

Â So what you can do is, we can actually replace the summation which runs from j

Â equals zero, to run from j equals let's call it etta, say.

Â Where etta is the minimum j, such that the stock price again this is St, after j up

Â moves and n minus j down-moves. So 8 is the minimum number of up-moves

Â required to ensure that the stock price, is greater than or equal to k.

Â In that case, this maximum will always occur in the first argument here.

Â So we can remove the max, remove the 0 And just substitute this expression in here,

Â inside the summation, and then we can split the summation up into two components

Â as we've done here. So, what you have is, we can rewrite, in

Â the binomial model, we can write c0, the initial value of the option, as being

Â equal to s0 times some quantity minus k times some quantity.

Â And if you recall, that's exactly what you had in our expression for the black

Â Black-Scholes we, we had s0 times some quantity, minus K times some quantity.

Â What you can do, and we won't do it, but you can show that if n goes to infinity,

Â or equivalently, if delta-t goes to 0, remember capital T is fixed, so if in goes

Â to infinity, delta-t goes to 0. You can show that if n goes to infinite,

Â Then c0, as we have here, will actually converge to the Black-Sholes formula.

Â Very briefly there is some great history associated with the modeling of Brownian

Â motion and Geometric Brownian motion, Stochastic calculus and finance in the

Â pricing of options. There are many famous names, both

Â mathematicians and economists, who are associated with this history, you might

Â want to take a look at some of these names in your spare time.

Â I'll just throw out a couple of very interesting people.

Â Bachelier, back in 1900, was perhaps the first to model Brownian motion, he was

Â attempting to do so, with a view to pricing options on the Paris stock

Â exchange. Another fascinating person, is Doeblin,

Â another French mathematician, whose work was only recently Discovered.

Â He was actually very involved in the development of Stochastic calculus, but

Â his work wasn't discovered until very recently, as he died in World War 2 before

Â his work could be read. Another very influential person or

Â fascinating person is Ed Thorp! Ed Thorp is famous for card counting and

Â showing that you can actually card count and beat the system in Las Vegas.

Â After he was thrown out of Vegas, apparently he started trading options and

Â may well have been the first person to actually discover and use the

Â Black-Scholes formula. He didn't actually prove it was true, he

Â didn't have a model which derived the Black-Scholes formula, but it seems like

Â he somehow intuited that, that might have been the correct formula for pricing

Â options. And there's a host of other interesting

Â names here, that are well worth exploring.

Â